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GramSchmidt.cc

// Copyright (C) 2002 Andreas Stahel
//
// This program is free software; you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation; either version 2 of the License, or
// (at your option) any later version.
//
// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with this program; If not, see <http://www.gnu.org/licenses/>.

#include <iostream>
#include <cmath>

#include <octave/oct.h>

#include <octave/parse.h>
#include <octave/pager.h>

#define max(a,b) (((a)<(b)) ? (b) : (a))
#define min(a,b) (((a)<(b)) ? (a) : (b))

//////////////////////////////////////////////////


void GramSchmidt(Matrix &V, ColumnVector &norms,int Vr, int Vc){
  double tmp=0.0;
  for(int i=0; i<Vc; i++){
    tmp=0.0;  // normalize column i
    for(int j=Vr-1;j>=0;j--){tmp += V(j,i)*V(j,i);}
    tmp = norms(i)= sqrt(tmp);
    for(int j=Vr-1;j>=0;j--){V(j,i)/=tmp;}
    for(int k=i+1;k<Vc;k++){
      tmp=0.0; // scalar product tmp=<V(:,i),V(:,k)>
      for(int kk=Vr-1;kk>=0;kk--) tmp += V(kk,i)*V(kk,k);
      // V(:,k) = V(:,k)-tmp*A(:,i)
      for(int kk=Vr-1;kk>=0;kk--) V(kk,k) -= tmp*V(kk,i);  
    };
  };
};

//////////////////////////////////////////////////


DEFUN_DLD (GramSchmidt, args, , "[...] = GramSchmidt(...)\n\
  apply the Gram Schmidt reduction to the columns of a matrix V\n\
\n\
  Vout = GramSchmidt(V)\n\
  [Vout, ColLength] = GramSchmidt(V)\n\
\n\
   V    is is a matrix of size mxn\n\
   Vout is is a matrix of size mxn,\n\
        the columns of Vout are orthonormalized and we have\n\
        span(V(:,1:k)) = span(Vout(:,1:k)) for k=1...n\n\
   ColLength is a vector containing the lengths of the column vectors of V\n\
        during the Gram Schmidt algorithm\n\
\n\
   The implementation is based of the modified Gram Schmidt algorithm as\n\
   described in \"Matrix Computations\" by G. Golub and C. van Loan")

{
  octave_value_list retval;
  
  int nargin = args.length ();
  if (nargin != 1) {
    print_usage ();
    return retval;
  }

  octave_value V_arg = args(0);
  int col = V_arg.columns();
  int row = V_arg.rows();
  Matrix V= V_arg.matrix_value();
  
  ColumnVector ColLength(col);
  
  GramSchmidt(V,ColLength,row,col);
  
  retval(0)= V;
  retval(1)= ColLength;
  return retval;
}

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